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a surprising source of mathematical problem-solving and teaching

What's this have to do with lectures and my students' complaints? Well, far too often, lectures and recitation sessions are just like the conversations Steve and I had with Paul. When somebody else is presenting a detailed explanation of how they solve some problem, it's very easy to nod along and say "Yes, yes, of course, that's the thing to do." You leave the room perfectly convinced that you've understood everything, but when you try to apply what you think you know by explaining it to someone else, you find that you didn't really understand a thing.

That's the problem with good explanations: they're incredibly seductive, convincing you that you understand things that you don't understand at all.

scientists from the University of Chicago reported that young children who understand how shapes fit together are better able to use a number line and to solve computation problems.

The theme here is fun math books. Thus, if a book says something like “this is a division sign,” you will not find the book on this list. These books get kids thinking about numbers and problem solving in neat ways without realizing that they are learning math.

I quickly figured out how to play that game successfully – success in that case being measured by my being able to solve under exam conditions, problems like the ones the teacher had shown us and we had practiced in class and done for homework.

In fact, you can’t separate real teaching from learning. They are simply two perspectives of the same human interactive process.

For whereas technology can provide instruction and can provide teachers and students with resources to assist them, what is cannot do on its own is teach them.